#243852

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,7,5,8,4,6)(9,10,14,18,15,19,13,28)(11,20,23,31,24,35,22,16)(12,25,26,33,27,17,29,34)(21,36,37,32,38,40,39,30);(2,6)(3,4)(7,8)(9,11,13,23,15,24,14,22)(10,16,28,20,19,31,18,35)(12,21,29,37,27,38,26,39)(17,32,33,40,25,30,34,36);(2,7)(3,4)(6,8)(9,12)(10,17)(11,21)(13,29)(14,26)(15,27)(16,30)(18,34)(19,25)(20,36)(22,39)(23,37)(24,38)(28,33)(31,32)(35,40);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,18,19,28)(11,22,24,23)(12,26,27,29)(16,20,31,35)(17,33,25,34)(21,37,38,39)(30,40,32,36)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151048);